picture 1:
I will remove the old dead grass (in that walk way) and extend the paving to the main lawn area.
You will have a hard time establishing lawn in the pictured area. Problems you might expierence there is traffic and if it is wet like I see in the picture it's even worse. Other problems there can be shade from the neighboring vegetation and the grass and soil dont dry off enough causing the grass to rot.
Picture 2:
Looks like a drainage problem (I see standing water puddles there) I'll do the same on the other end of that gate too. Get it graded so water will run off and put down a few more of those slate stones. Seed inbetween them.
Good Luck
PC

Your plan is great. I think anything you try or any effort can only improve the situation.
I dont know up in Canada but around here the guys will aerate and seed or interseed. We will also put a little bit of starter fertilizer (13-13-13) down on the newly seeded or sodded areas. Have you considered sodding those areas?
PC